Ratios and Proportions Alıştırma Soruları — Armed Services Vocational Aptitude Battery (ASVAB) — Sayfa 5

Fill in the blank with the correct numeric value to complete the statement.

Aşağıdaki boşlukları doldurun

A printing press can print

50

identical banners in

2

minutes. At this constant rate, the press will print banners in

10

minutes.

Cevabı ve açıklamayı göster

Cevap

250

Because the printing press operates at a constant rate, the number of banners produced is directly proportional to the time spent printing. Since

10

minutes is exactly

5

times longer than the initial

2

minutes, the press will produce

5

times as many banners. Multiplying the original

50

banners by

5

results in

250

banners.

Adım Adım Çözüm

1

Identify the given ratio of banners to time.

The press produces

50

banners per

2

minutes.

Establishing the baseline rate is the first step in setting up a direct proportion.

2

Determine the scaling factor for the time.

The time increases from

2

minutes to

10

minutes. The scaling factor is

\frac{10}{2} = 5

.

Finding how many times greater the new time is compared to the original time tells us how to scale the output.

3

Multiply the baseline production by the scaling factor.

50 \times 5 = 250

banners.

In a direct proportion, as time increases by a certain multiple, the quantity produced increases by that exact same multiple.

Anahtar Kavram

Direct Proportions

Soru 82Soru →

A supply pump transfers $10$ gallons of aviation fuel every $15$ seconds. At this rate, how many gallons of fuel will it transfer in $3$ minutes?

120

270

12

2

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

120

The problem establishes a constant rate, which requires a direct proportion to solve. First, convert the

3

-minute time period into

180

seconds to match the units of the given rate. Then, set up the proportion

\frac{10}{15} = \frac{x}{180}

. Cross-multiplying

10

by

180

yields

1800

, and dividing that product by

15

results in

120

gallons.

Adım Adım Çözüm

1

Harmonize the time units by converting minutes to seconds.

3

minutes

= 3 \times 60 = 180

seconds.

The initial rate is given in seconds, so the target time must also be in seconds to set up an accurate mathematical proportion.

2

Set up a direct proportion using the given rate.

10

gallons /

15

seconds

= x

gallons /

180

seconds.

The pump operates at a constant rate, meaning the ratio of volume to time remains equivalent.

3

Solve for the unknown volume

x

.

x = \frac{10 \times 180}{15} = \frac{1800}{15} = 120

gallons.

Cross-multiplying and dividing isolates the variable, providing the total gallons transferred.

Anahtar Kavram

Direct proportions represent situations where two quantities increase at the same relative rate. When setting up proportions, it is critical to ensure that corresponding units match.

Soru 83Soru →

An industrial cleaning solvent requires a mixture of Chemical A and Chemical B in a ratio of $5:2$ by volume. If a worker uses $300$ milliliters of Chemical B, how many liters of the final cleaning solvent will be produced?

1050

0.75

1.05

10.5

Cevabı ve açıklamayı göster

Cevap: 1.05

Cevap

1.05

The correct answer accurately scales the mixture from parts to a total volume. Since the ratio of Chemical A to B is 5:2, there are 7 total parts in the mixture. Chemical B makes up 2 of these 7 parts. Setting up the proportion 2/7 = 300/x gives a total volume of 1050 mL. Converting this volume to liters by dividing by 1000 results in 1.05 liters.

Adım Adım Çözüm

1

Identify the ratio of Chemical B to the total mixture.

The mixture has 5 parts Chemical A and 2 parts Chemical B, making 7 total parts. The ratio of Chemical B to the total is 2:7.

To find the total volume based on the volume of one component, we need the part-to-whole relationship.

2

Set up and solve a proportion to find the total volume in milliliters.

2 / 7 = 300 / x. Cross-multiplying gives 2x = 2100, which simplifies to x = 1050 milliliters.

Direct proportions allow us to scale the ratio up to find the actual combined volume of the mixture.

3

Convert the total volume from milliliters to liters.

1050 / 1000 = 1.05 liters.

The question explicitly requires the final answer to be stated in liters.

Anahtar Kavram

Direct proportions and unit conversions in part-to-whole relationships

Alternatif Yöntem

Instead of setting up a formal proportion, use the unitary method. If 2 parts represent 300 mL, then 1 part equals 150 mL (300 / 2). The total mixture requires 7 parts, so the total volume is 7 * 150 = 1050 mL. Converting to liters gives 1.05 L.

Tahmini Süre:1m 30s

Soru 84Soru →

Solve the following word problem by calculating the missing value.

Aşağıdaki boşlukları doldurun

An engineering battalion can lay

40

feet of runway matting in

15

minutes. Operating at this constant rate, the battalion will lay feet of matting in

1.5

hours.

Cevabı ve açıklamayı göster

Cevap

240

Because

1.5

hours is equivalent to

90

minutes, the time period is exactly

6

times longer than the original

15

-minute period (

90 \div 15 = 6

). In a direct proportion, the amount of matting laid increases by the same factor, so

40 \text{ feet} \times 6 = 240 \text{ feet}

.

Adım Adım Çözüm

1

Convert the target time frame into the same units used for the known rate.

1.5 \text{ hours} = 1.5 \times 60 \text{ minutes} = 90 \text{ minutes}

Units of time must match to set up a valid direct proportion.

2

Set up a proportion relating feet of matting to minutes.

\frac{40 \text{ feet}}{15 \text{ minutes}} = \frac{x \text{ feet}}{90 \text{ minutes}}

Because the rate is constant, the ratio of distance to time remains the same.

3

Solve for the unknown value

x

.

x = \frac{40 \times 90}{15} = 40 \times 6 = 240

Isolating

x

provides the total feet of matting laid in

90

minutes.

Anahtar Kavram

Direct proportions with unit conversion

Soru 85Soru →

A maintenance crew can paint $120$ square feet of a ship's hull in $2$ hours. Working at this same constant rate, how many square feet can the crew paint in $5$ hours?

Cevabı ve açıklamayı göster

Cevap: 300

Cevap

300

Because the work happens at a constant rate, the area painted is directly proportional to the time spent working. Finding the unit rate (

60

square feet per hour) and multiplying it by the target time (

5

hours) yields the correct total of

300

square feet.

Adım Adım Çözüm

1

Calculate the unit rate of painting.

60

square feet per hour

To find how much is painted in a different amount of time, we first need to know how much is painted in one hour. We do this by dividing the total area (

120

) by the total time (

2

).

2

Calculate the total area painted in the new timeframe.

300

square feet

Multiply the unit rate (

60

square feet per hour) by the new target time (

5

hours) to find the final proportional amount.

Anahtar Kavram

Direct Proportions

Soru 86Soru →

A commercial bakery makes a standard pastry dough using a ratio of $7$ parts flour to $3$ parts butter by weight. If a daily batch requires $42$ pounds of flour, how many pounds of butter are needed?

18

60

98

126

Cevabı ve açıklamayı göster

Cevap: $18$

Cevap

18

The ratio of flour to butter is given as

7:3

. Setting up a direct proportion where

x

is the pounds of butter needed:

\frac{7}{3} = \frac{42}{x}

. Cross-multiplying gives

7x = 126

. Solving for

x

yields

x = 18

pounds of butter.

Adım Adım Çözüm

1

Identify the given ratio and the known quantity.

The ratio of flour to butter is

7:3

. The known quantity of flour is

42

pounds.

To set up a proportion, we must align the correct parts of the ratio with the real-world amounts.

2

Set up a direct proportion equation.

\frac{7}{3} = \frac{42}{x}

The ratio of parts must equal the ratio of the actual weights.

3

Solve for the unknown value

x

.

7x = 3 \times 42 \rightarrow 7x = 126 \rightarrow x = 18

Cross-multiplying and dividing isolates the variable to find the required pounds of butter.

Anahtar Kavram

Direct Proportions

Soru 87Soru →

An automated wire-cutting machine processes $350$ meters of copper wiring in $20$ minutes. Operating at this constant rate, how many meters of wiring will the machine process in $1.5$ hours?

Cevabı ve açıklamayı göster

Cevap: 1575

Cevap

1575

Because the machine operates at a constant rate, the relationship between time and wire processed is directly proportional. However, the time units differ between the given rate (

20

minutes) and the target time (

1.5

hours). First, convert

1.5

hours to

90

minutes. Then, set up the direct proportion

\frac{350 \text{ meters}}{20 \text{ minutes}} = \frac{x \text{ meters}}{90 \text{ minutes}}

. Solving for

x

by multiplying

350

by

90

and dividing by

20

yields exactly

1575

meters.

Adım Adım Çözüm

1

Harmonize the time units by converting hours to minutes.

90

minutes

The given rate is in meters per minute, so the target time must also be in minutes to establish a valid equivalent ratio.

2

Set up a direct proportion equation.

\frac{350}{20} = \frac{x}{90}

The length of wire processed increases at a constant, directly proportional rate to the time operated.

3

Solve the proportion for the unknown value

x

.

x = 1575

Cross-multiplying and simplifying isolates

x

, calculating the total meters processed in the longer timeframe.

Anahtar Kavram

Direct proportions with unit harmonization

Soru 88Soru →

A forward operating base maintains a stockpile of ammunition, food, and medical supplies in a weight ratio of 5:3:2. An incoming supply convoy delivers an additional 3 tons of food and 8 tons of medical supplies, while the ammunition stockpile remains unchanged. After the delivery, the new weight ratio of food to medical supplies becomes 4:3. What was the total weight of the entire stockpile before the convoy arrived?

20 tons

23 tons

115 tons

230 tons

Cevabı ve açıklamayı göster

Cevap: 230 tons

Cevap

The total weight of the entire stockpile before the convoy arrived was 230 tons.

The original weights can be expressed proportionally as

5x

(ammunition),

3x

(food), and

2x

(medical supplies), totaling

10x

. After adding the newly delivered amounts, the new weights for food and medical supplies become

3x + 3

and

2x + 8

. Setting their ratio equal to 4:3 yields the equation

\frac{3x + 3}{2x + 8} = \frac{4}{3}

. Cross-multiplying gives

9x + 9 = 8x + 32

, which simplifies to

x = 23

. The total original weight is therefore

10x = 10(23) = 230

tons.

Adım Adım Çözüm

1

Define the original weights of the supplies using a common multiplier

x

.

Ammunition =

5x

, Food =

3x

, Medical =

2x

. Total original weight =

5x + 3x + 2x = 10x

.

A ratio represents proportional parts of a whole. Multiplying each ratio term by a constant

x

gives expressions for their actual quantities.

2

Set up an algebraic expression for the new weights of food and medical supplies after the delivery.

New Food =

3x + 3

; New Medical =

2x + 8

.

The convoy delivered 3 tons of food and 8 tons of medical supplies, which are added to their respective original expressions.

3

Create a proportion using the new ratio of food to medical supplies provided in the problem.

\frac{3x + 3}{2x + 8} = \frac{4}{3}

The problem states that after the delivery, the ratio of food to medical supplies becomes 4:3.

4

Solve the proportion for

x

using cross-multiplication.

3(3x + 3) = 4(2x + 8) \Rightarrow 9x + 9 = 8x + 32 \Rightarrow x = 23

.

Cross-multiplying eliminates the fractions, creating a linear equation that can be solved for the common multiplier

x

.

5

Calculate the total original weight of the stockpile by substituting the value of

x

.

10(23) = 230

tons.

The question asks for the total weight of the entire stockpile before the delivery, which is represented by the expression

10x

.

Anahtar Kavram

Setting up and solving algebraic proportions from multi-term ratios after quantities undergo additive changes.

Tahmini Süre:1m 30s

Soru 89Soru →

A field water purification system processes $12.5$ gallons of clean water every $5$ minutes. Operating at this same constant rate, how many gallons of clean water will the system process in $2$ hours?

5

30

48

300

Cevabı ve açıklamayı göster

Cevap: 300

Cevap

The system will process 300 gallons of clean water in 2 hours.

The correct answer accurately harmonizes the time units by converting 2 hours into 120 minutes, and then scales the given volume-to-time ratio (

12.5

gallons per

5

minutes) up to

120

minutes, resulting in

300

gallons.

Adım Adım Çözüm

1

Convert the target time into the same units used in the given rate.

2

hours

= 2 \times 60 = 120

minutes.

Proportions require matching units to accurately compare the two ratios.

2

Calculate the unit rate of water processed per minute.

12.5 \text{ gallons} \div 5 \text{ minutes} = 2.5 \text{ gallons per minute}

.

Finding the amount processed in one minute simplifies scaling up to the total time.

3

Multiply the unit rate by the total number of minutes.

2.5 \text{ gallons/minute} \times 120 \text{ minutes} = 300 \text{ gallons}

.

This applies the constant rate over the entire duration to find the total volume.

Anahtar Kavram

Direct proportions with unit conversion

Soru 90Soru →

A specialized construction $3$ D printer uses $2.4$ metric tons of concrete to print $15$ linear feet of a wall. At this constant rate, how many kilograms of concrete are required to print a wall that is $25$ yards long? ( $1$ metric ton = $1,000$ kilograms; $1$ yard = $3$ feet)

4,000

1,200

12,000

480

Cevabı ve açıklamayı göster

Cevap: $12,000$

Cevap

The correct answer is

12,000

.

The correct answer is found by harmonizing the length units, determining the scaled proportion, and harmonizing the final weight units. Converting

25

yards to

75

feet reveals that the target length is

5

times the initial length (

75 \div 15 = 5

). Multiplying the initial concrete requirement by this scale factor yields

12

metric tons (

2.4 \times 5 = 12

). Finally, converting

12

metric tons to kilograms yields

12,000

kilograms.

Adım Adım Çözüm

1

Convert the target length from yards to feet.

25 \text{ yards} = 25 \times 3 = 75 \text{ feet}

.

The units of length must match the given rate (

15

linear feet) before setting up a proportion.

2

Set up a direct proportion to find the metric tons of concrete needed.

\frac{2.4}{15} = \frac{x}{75}

The amount of concrete used is directly proportional to the length of the wall printed.

3

Solve the proportion for

x

.

x = \frac{2.4 \times 75}{15} = 2.4 \times 5 = 12 \text{ metric tons}

.

Isolating

x

provides the required amount of concrete in metric tons.

4

Convert metric tons to kilograms.

12 \text{ metric tons} = 12 \times 1,000 = 12,000 \text{ kilograms}

.

The question specifically asks for the final answer in kilograms.

Anahtar Kavram

Direct Proportions with Unit Conversions

Soru 91Soru →

A military water purification unit has three filters operating at different unit rates. Filter X can purify $180$ gallons of water in $1.5$ hours. Filter Y can purify $320$ gallons in $2$ hours. Filter Z's purification rate is exactly $20\%$ faster than Filter X's rate. If all three filters are turned on at the same time, how many minutes will it take them to completely purify $1,484$ gallons of water?

Cevabı ve açıklamayı göster

Cevap: 210

Cevap

210

The correct answer is found by first determining the individual hourly rates of each filter, adding them together for a combined rate, calculating the total hours required for the given volume, and finally converting that time into minutes.

Adım Adım Çözüm

1

Calculate the unit rate for Filter X.

120

gallons per hour

To find the combined rate later, we first need the individual hourly rate of Filter X.

180 \div 1.5 = 120

.

2

Calculate the unit rate for Filter Y.

160

gallons per hour

We need the individual hourly rate of Filter Y to find the total combined rate.

320 \div 2 = 160

.

3

Calculate the unit rate for Filter Z.

144

gallons per hour

Filter Z operates

20\%

faster than Filter X, so its rate is

120 + (0.20 \times 120) = 144

.

4

Calculate the combined unit rate of all three filters.

424

gallons per hour

Since the filters operate simultaneously, their individual rates add together:

120 + 160 + 144 = 424

.

5

Determine the time in hours needed to purify

1,484

gallons.

3.5

hours

Time equals the total amount required divided by the combined rate:

1,484 \div 424 = 3.5

.

6

Convert the total time from hours to minutes.

210

minutes

The question explicitly asks for the answer in minutes.

3.5 \times 60 = 210

.

Anahtar Kavram

Calculating, modifying, and combining multiple unit rates, followed by a final unit conversion.

Alternatif Yöntem

You could convert the rates to gallons per minute immediately. Filter X is 2 gallons per minute, Filter Y is 2.67 gallons per minute, etc. However, this introduces repeating decimals early on, making the math significantly harder without a calculator. Keeping the rates in hours avoids complex fractions until the final step.

Tahmini Süre:3m 30s

Soru 92Soru →

An earth-moving excavator clears $120$ cubic yards of debris in $45$ minutes. Operating at this constant rate, how many cubic yards of debris will the excavator clear in $3$ hours?

480

8

30

48

Cevabı ve açıklamayı göster

Cevap: 480

Cevap

480

The correct answer is found by first converting

3

hours into

180

minutes. Then, applying a direct proportion,

\frac{120}{45} = \frac{x}{180}

, yields

x = 480

. Alternatively, observing that

180

minutes is exactly

4

times as long as

45

minutes means the excavator will clear

4

times the debris:

120 \times 4 = 480

.

Adım Adım Çözüm

1

Convert the target time from hours to minutes so the units match the given rate.

3 \text{ hours} \times 60 \text{ minutes/hour} = 180 \text{ minutes}

Units of time must be identical before setting up a valid proportion.

2

Set up a direct proportion relating the volume of debris to the time taken.

\frac{120}{45} = \frac{x}{180}

Since the excavator works at a constant rate, the ratio of volume to time remains constant.

3

Solve for the unknown volume (

x

) by cross-multiplying or finding a common multiplier.

x = \frac{120 \times 180}{45} = 120 \times 4 = 480

Isolating

x

provides the total cubic yards cleared in

180

minutes.

Anahtar Kavram

Direct proportions with unit conversions

Tahmini Süre:1m 0s

Soru 93Soru →

Determine the correct numerical value to complete the statement below based on the provided rates.

Aşağıdaki boşlukları doldurun

An airborne logistics division drops

3.5

tons of humanitarian rations every

12

minutes. Operating at this constant rate, the division will drop pounds of rations in

1.5

hours.

*(Note:

1 \text{ ton} = 2,000 \text{ pounds}

)*

Cevabı ve açıklamayı göster

Cevap

52,500

The scenario requires calculating a proportional amount while successfully navigating two unit conversions. First, harmonize the time units by converting

1.5

hours to

90

minutes. Next, convert the initial weight of

3.5

tons into

7,000

pounds. Using these standardized units, set up the proportion:

7,000 \text{ lbs} / 12 \text{ min} = x \text{ lbs} / 90 \text{ min}

. Cross-multiplying gives

12x = 630,000

, and dividing by

12

results in exactly

52,500

pounds.

Adım Adım Çözüm

1

Convert the target time frame from hours to minutes so it matches the time unit in the given rate.

1.5 \text{ hours} \times 60 \text{ minutes/hour} = 90 \text{ minutes}

Direct proportions require time units to be identical across both ratios.

2

Convert the initial weight from tons to pounds to match the requested unit for the final answer.

3.5 \text{ tons} \times 2,000 \text{ pounds/ton} = 7,000 \text{ pounds}

Harmonizing the weight units prevents conversion errors at the end of the calculation.

3

Set up a direct proportion equation comparing weight to time.

\frac{7,000 \text{ pounds}}{12 \text{ minutes}} = \frac{x \text{ pounds}}{90 \text{ minutes}}

Because the drop rate is constant, the ratio of weight to time remains equal.

4

Solve for the unknown weight (x).

x = \frac{7,000 \times 90}{12} = \frac{630,000}{12} = 52,500 \text{ pounds}

Cross-multiplying and dividing isolates the variable, yielding the total pounds dropped over the full time period.

Anahtar Kavram

Setting up and solving a direct proportion after performing multiple unit conversions.

Soru 94Soru →

Read the scenario and fill in the blank with the correct numerical value.

Aşağıdaki boşlukları doldurun

A logistics team can pack

30

ration boxes in

15

minutes. Operating at this same constant rate, the team can pack ration boxes in

45

minutes.

Cevabı ve açıklamayı göster

Cevap

90

Because the packing rate is constant, the relationship between boxes and minutes is directly proportional. Determining the unit rate (

2

boxes per minute) and multiplying it by the target time (

45

minutes) correctly yields

90

boxes.

Adım Adım Çözüm

1

Calculate the unit rate of packing.

30 \div 15 = 2

boxes per minute.

To find out how many boxes can be packed in a different amount of time, it helps to first know how many boxes are packed in exactly one minute.

2

Multiply the unit rate by the new time.

2 \times 45 = 90

boxes.

Since the team packs 2 boxes every minute, multiplying this constant rate by 45 minutes gives the total number of boxes packed in that timeframe.

Anahtar Kavram

Direct Proportions

Alternatif Yöntem

You can also solve this by setting up a direct proportion equation:

\frac{30}{15} = \frac{x}{45}

. Recognizing that

45

is exactly

3

times

15

, you can simply multiply the initial number of boxes by

3

:

30 \times 3 = 90

.

Tahmini Süre:45s

Soru 95Soru →

An industrial textile loom weaves $4.5$ yards of heavy canvas every $12$ minutes. Operating at this same constant rate, how many feet of canvas will the loom produce during a complete $8$ -hour work shift?

Cevabı ve açıklamayı göster

Cevap: 540

Cevap

540 feet

To solve this problem, you must properly scale a direct proportion while paying careful attention to units. First, harmonize the time units: an 8-hour shift is equal to 480 minutes (8 × 60). Next, set up a proportion or find the scale factor: 480 minutes is exactly 40 times longer than the 12-minute baseline (480 ÷ 12 = 40). Because the loom operates at a constant rate, it will produce 40 times as much canvas. Multiply the 4.5 yards by 40 to find the total yards produced (4.5 × 40 = 180 yards). Finally, the question asks for the answer in feet. Since there are 3 feet in 1 yard, multiply 180 yards by 3 to get exactly 540 feet.

Adım Adım Çözüm

1

Convert the total operating time from hours to minutes to match the given rate's unit of time.

8 \text{ hours} \times 60 \text{ minutes/hour} = 480 \text{ minutes}

Units of time must be identical before calculating the direct proportion.

2

Determine how many 12-minute intervals occur in the 480-minute shift.

480 \div 12 = 40 \text{ intervals}

This establishes the multiplier (scale factor) that will be applied to the fabric production.

3

Calculate the total yards of canvas produced during the entire shift.

40 \times 4.5 \text{ yards} = 180 \text{ yards}

Because production is directly proportional to time, multiplying the base production by the time scale factor yields total yards.

4

Convert the total production from yards to feet to answer the specific question asked.

180 \text{ yards} \times 3 \text{ feet/yard} = 540 \text{ feet}

The stem explicitly asks for the final length in feet, making this unit conversion a critical final step.

Anahtar Kavram

Solving multi-step direct proportion problems that require initial and final unit harmonizations (time and length).

Soru 96Soru →

A portable generator consumes $8$ liters of fuel every $12$ hours of operation. At this constant rate, how many liters of fuel will the generator consume in $48$ hours of operation?

2

3.2

32

72

Cevabı ve açıklamayı göster

Cevap: 32

Cevap

The generator will consume 32 liters of fuel.

Because the generator operates at a constant rate, the relationship between time and fuel is a direct proportion. The new operating time (

48

hours) is exactly

4

times longer than the original operating time (

12

hours). Therefore, the fuel consumption will also be

4

times greater:

8 \text{ liters} \times 4 = 32 \text{ liters}

.

Adım Adım Çözüm

1

Determine the relationship between the two operating times.

48 \div 12 = 4

Finding the scale factor makes it simple to calculate the proportionally larger amount of fuel.

2

Multiply the original fuel amount by the scale factor.

8 \times 4 = 32

liters

In a direct proportion, whatever factor scales the time must also scale the fuel consumption.

Anahtar Kavram

Direct proportions state that as one quantity increases by a certain multiplier, the related quantity increases by that exact same multiplier.

Soru 97Soru →

Complete the statement below by calculating the missing numerical value. Enter only the number.

Aşağıdaki boşlukları doldurun

A military satellite uplink transmits

1.5

gigabytes of reconnaissance data every

40

seconds. Operating at this constant rate, it will take minutes to transmit a total of

18

gigabytes of data.

Cevabı ve açıklamayı göster

Cevap

8

By setting up a proportion, we find that transmitting

18

gigabytes takes

480

seconds. Dividing

480

seconds by

60

gives exactly

8

minutes.

Adım Adım Çözüm

1

Set up a direct proportion comparing data volume to time.

\frac{1.5}{40} = \frac{18}{x}

Because the transmission rate is constant, the ratio of gigabytes to seconds remains the same.

2

Solve for the unknown time

x

in seconds by cross-multiplying.

1.5x = 40 \times 18 \rightarrow 1.5x = 720 \rightarrow x = 480

seconds

Cross-multiplication isolates the variable, giving the total time required in seconds.

3

Convert the total seconds into minutes.

480 \div 60 = 8

minutes

The question specifically asks for the time in minutes, so the time in seconds must be divided by the number of seconds in a minute (

60

).

Anahtar Kavram

Direct proportions with unit conversions

Tahmini Süre:1m 30s

Soru 98Soru →

A naval vessel's desalination plant produces fresh water at a rate directly proportional to its operational time. The plant produces $1,500$ liters of fresh water in $40$ minutes. The vessel's main storage tank has a total capacity of $24$ cubic meters and is currently $25\%$ full.

Assuming $1$ cubic meter equals $1,000$ liters, determine the correct numerical value to answer the question: How many continuous hours must the desalination plant operate to completely fill the remainder of the storage tank?

Aşağıdaki boşlukları doldurun

The plant must operate for hours.

Cevabı ve açıklamayı göster

Cevap

8

The correct answer is 8 hours. First, convert the tank capacity to liters:

24 \text{ m}^3 = 24,000 \text{ L}

. Since the tank is

25\%

full, it needs

75\%

of its capacity to be filled, which is

24,000 \times 0.75 = 18,000 \text{ L}

. Using the direct proportion

\frac{40}{1,500} = \frac{x}{18,000}

, we find

x = 480

minutes. Finally, converting minutes to hours by dividing by

60

gives exactly

8

hours.

Adım Adım Çözüm

1

Calculate the total tank capacity in liters.

24 \times 1,000 = 24,000

liters.

To harmonize the tank's volume units with the desalination plant's production rate, which is given in liters.

2

Determine the volume of water needed to fill the remainder of the tank.

24,000 \times (1 - 0.25) = 18,000

liters.

The tank is already

25\%

full, meaning only the remaining

75\%

needs to be produced.

3

Set up and solve a direct proportion to find the required time in minutes.

\frac{40 \text{ min}}{1,500 \text{ L}} = \frac{x \text{ min}}{18,000 \text{ L}} \Rightarrow x = \frac{40 \times 18,000}{1,500} = 480

minutes.

Because water production is directly proportional to operational time.

4

Convert the required time from minutes to hours.

480 \div 60 = 8

hours.

The question specifically asks for the operational time in continuous hours.

Anahtar Kavram

Solving multi-step direct proportions involving unit harmonization and percentages.

Soru 99Soru →

A naval ship uses $400$ pounds of provisions to feed a specific crew for $5$ days. At this same constant rate, how many pounds of provisions are needed to feed the crew for $12$ days?

Cevabı ve açıklamayı göster

Cevap: 960

Cevap

960

By determining the unit rate of

80

pounds per day and multiplying it by

12

days, the direct proportionality rule is correctly applied to find the total of

960

pounds.

Adım Adım Çözüm

1

Calculate the unit rate of provisions per day.

400 \div 5 = 80

pounds per day.

Finding the amount used in one day allows scaling the proportion to any other number of days.

2

Multiply the unit rate by the new number of days.

80 \times 12 = 960

pounds.

This scales the daily rate up to the required

12

-day period to maintain the direct proportion.

Anahtar Kavram

Direct proportions using unit rates

Tahmini Süre:45s

Soru 100Soru →

A military transport aircraft consumes $3,600$ pounds of aviation fuel during a $45$ -minute flight at cruising altitude. Operating at this same constant rate of fuel consumption, how many tons of fuel will the aircraft consume during a $6$ -hour flight at cruising altitude? ( $1 \text{ ton} = 2,000 \text{ pounds}$ )

8.1

14.4

24.0

10.8

Cevabı ve açıklamayı göster

Cevap: $14.4$

Cevap

The aircraft will consume

14.4

tons of fuel.

Fuel consumption is directly proportional to flight time. First, harmonize the time units by finding the scale factor between the two flights. A

6

-hour flight is

360

minutes. The proportion multiplier is

360 \div 45 = 8

. Multiplying the original

3,600

pounds by

8

gives

28,800

pounds for the longer flight. Finally, convert this weight to tons by dividing by

2,000

, resulting in

14.4

tons.

Adım Adım Çözüm

1

Calculate the scale factor between the target flight time and the initial flight time.

The target time is

6

hours, which equals

360

minutes (

6 \times 60

). The initial time is

45

minutes. The scale factor is

360 \div 45 = 8

.

Finding the multiplier makes it easy to scale up the fuel consumption proportionally.

2

Calculate the total fuel consumed in pounds for the

6

-hour flight.

3,600 \text{ pounds} \times 8 = 28,800 \text{ pounds}

.

Since fuel consumption is directly proportional to flight time, multiplying the initial fuel by the time scale factor gives the total fuel.

3

Convert the total fuel from pounds to tons.

28,800 \div 2,000 = 14.4 \text{ tons}

.

The question asks for the final answer in tons, and we are given that

1 \text{ ton} = 2,000 \text{ pounds}

.

Anahtar Kavram

Direct Proportions with Unit Conversions

Alternatif Yöntem

Instead of finding the scale factor, you can find the unit rate per minute:

3,600 \div 45 = 80

pounds per minute. Then, convert the

6

-hour flight to minutes (

6 \times 60 = 360

minutes). Multiply the rate by the total minutes (

80 \times 360 = 28,800

pounds), and finally convert to tons (

28,800 \div 2,000 = 14.4

tons).

Tahmini Süre:2m 0s

Ratios and Proportions

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